Average Error: 0.2 → 0.2
Time: 25.3s
Precision: binary64
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \]
↓
\[\begin{array}{l} t_1 := {\sin \sin^{-1} \left(\mathsf{fma}\left(\cos delta, \sin \phi_1, \sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right)\right)}^{3} \cdot {\sin \phi_1}^{3}\\ t_2 := \sqrt[3]{t_1}\\ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\sqrt[3]{\frac{{\left({\cos delta}^{3} - t_1\right)}^{3}}{{\left(\mathsf{fma}\left(t_2, \cos delta + t_2, {\cos delta}^{2}\right)\right)}^{3}}}} \end{array} \]
\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)}
↓
\begin{array}{l}
t_1 := {\sin \sin^{-1} \left(\mathsf{fma}\left(\cos delta, \sin \phi_1, \sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right)\right)}^{3} \cdot {\sin \phi_1}^{3}\\
t_2 := \sqrt[3]{t_1}\\
\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\sqrt[3]{\frac{{\left({\cos delta}^{3} - t_1\right)}^{3}}{{\left(\mathsf{fma}\left(t_2, \cos delta + t_2, {\cos delta}^{2}\right)\right)}^{3}}}}
\end{array}

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (+
  lambda1
  (atan2
   (* (* (sin theta) (sin delta)) (cos phi1))
   (-
    (cos delta)
    (*
     (sin phi1)
     (sin
      (asin
       (+
        (* (sin phi1) (cos delta))
        (* (* (cos phi1) (sin delta)) (cos theta))))))))))
↓
(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1
         (*
          (pow
           (sin
            (asin
             (fma
              (cos delta)
              (sin phi1)
              (* (sin delta) (* (cos phi1) (cos theta))))))
           3.0)
          (pow (sin phi1) 3.0)))
        (t_2 (cbrt t_1)))
   (+
    lambda1
    (atan2
     (* (* (sin theta) (sin delta)) (cos phi1))
     (cbrt
      (/
       (pow (- (pow (cos delta) 3.0) t_1) 3.0)
       (pow (fma t_2 (+ (cos delta) t_2) (pow (cos delta) 2.0)) 3.0)))))))
double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return lambda1 + atan2(((sin(theta) * sin(delta)) * cos(phi1)), (cos(delta) - (sin(phi1) * sin(asin((sin(phi1) * cos(delta)) + ((cos(phi1) * sin(delta)) * cos(theta)))))));
}
↓
double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = pow(sin(asin(fma(cos(delta), sin(phi1), (sin(delta) * (cos(phi1) * cos(theta)))))), 3.0) * pow(sin(phi1), 3.0);
	double t_2 = cbrt(t_1);
	return lambda1 + atan2(((sin(theta) * sin(delta)) * cos(phi1)), cbrt(pow((pow(cos(delta), 3.0) - t_1), 3.0) / pow(fma(t_2, (cos(delta) + t_2), pow(cos(delta), 2.0)), 3.0)));
}

Error

Bits error versus lambda1

Bits error versus phi1

Bits error versus phi2

Bits error versus delta

Bits error versus theta

Derivation

  1. Initial program 0.2

    \[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \]

  2. Simplified0.2

    \[\leadsto \color{blue}{\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\mathsf{fma}\left(\cos delta, \sin \phi_1, \left(\sin delta \cdot \cos \phi_1\right) \cdot \cos theta\right)\right)}} \]

  3. Applied add-cbrt-cube_binary640.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \color{blue}{\sqrt[3]{\left(\sin \sin^{-1} \left(\mathsf{fma}\left(\cos delta, \sin \phi_1, \left(\sin delta \cdot \cos \phi_1\right) \cdot \cos theta\right)\right) \cdot \sin \sin^{-1} \left(\mathsf{fma}\left(\cos delta, \sin \phi_1, \left(\sin delta \cdot \cos \phi_1\right) \cdot \cos theta\right)\right)\right) \cdot \sin \sin^{-1} \left(\mathsf{fma}\left(\cos delta, \sin \phi_1, \left(\sin delta \cdot \cos \phi_1\right) \cdot \cos theta\right)\right)}}} \]

  4. Applied add-cbrt-cube_binary640.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \color{blue}{\sqrt[3]{\left(\sin \phi_1 \cdot \sin \phi_1\right) \cdot \sin \phi_1}} \cdot \sqrt[3]{\left(\sin \sin^{-1} \left(\mathsf{fma}\left(\cos delta, \sin \phi_1, \left(\sin delta \cdot \cos \phi_1\right) \cdot \cos theta\right)\right) \cdot \sin \sin^{-1} \left(\mathsf{fma}\left(\cos delta, \sin \phi_1, \left(\sin delta \cdot \cos \phi_1\right) \cdot \cos theta\right)\right)\right) \cdot \sin \sin^{-1} \left(\mathsf{fma}\left(\cos delta, \sin \phi_1, \left(\sin delta \cdot \cos \phi_1\right) \cdot \cos theta\right)\right)}} \]

  5. Applied cbrt-unprod_binary640.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \color{blue}{\sqrt[3]{\left(\left(\sin \phi_1 \cdot \sin \phi_1\right) \cdot \sin \phi_1\right) \cdot \left(\left(\sin \sin^{-1} \left(\mathsf{fma}\left(\cos delta, \sin \phi_1, \left(\sin delta \cdot \cos \phi_1\right) \cdot \cos theta\right)\right) \cdot \sin \sin^{-1} \left(\mathsf{fma}\left(\cos delta, \sin \phi_1, \left(\sin delta \cdot \cos \phi_1\right) \cdot \cos theta\right)\right)\right) \cdot \sin \sin^{-1} \left(\mathsf{fma}\left(\cos delta, \sin \phi_1, \left(\sin delta \cdot \cos \phi_1\right) \cdot \cos theta\right)\right)\right)}}} \]

  6. Applied add-cbrt-cube_binary640.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\sqrt[3]{\left(\left(\cos delta - \sqrt[3]{\left(\left(\sin \phi_1 \cdot \sin \phi_1\right) \cdot \sin \phi_1\right) \cdot \left(\left(\sin \sin^{-1} \left(\mathsf{fma}\left(\cos delta, \sin \phi_1, \left(\sin delta \cdot \cos \phi_1\right) \cdot \cos theta\right)\right) \cdot \sin \sin^{-1} \left(\mathsf{fma}\left(\cos delta, \sin \phi_1, \left(\sin delta \cdot \cos \phi_1\right) \cdot \cos theta\right)\right)\right) \cdot \sin \sin^{-1} \left(\mathsf{fma}\left(\cos delta, \sin \phi_1, \left(\sin delta \cdot \cos \phi_1\right) \cdot \cos theta\right)\right)\right)}\right) \cdot \left(\cos delta - \sqrt[3]{\left(\left(\sin \phi_1 \cdot \sin \phi_1\right) \cdot \sin \phi_1\right) \cdot \left(\left(\sin \sin^{-1} \left(\mathsf{fma}\left(\cos delta, \sin \phi_1, \left(\sin delta \cdot \cos \phi_1\right) \cdot \cos theta\right)\right) \cdot \sin \sin^{-1} \left(\mathsf{fma}\left(\cos delta, \sin \phi_1, \left(\sin delta \cdot \cos \phi_1\right) \cdot \cos theta\right)\right)\right) \cdot \sin \sin^{-1} \left(\mathsf{fma}\left(\cos delta, \sin \phi_1, \left(\sin delta \cdot \cos \phi_1\right) \cdot \cos theta\right)\right)\right)}\right)\right) \cdot \left(\cos delta - \sqrt[3]{\left(\left(\sin \phi_1 \cdot \sin \phi_1\right) \cdot \sin \phi_1\right) \cdot \left(\left(\sin \sin^{-1} \left(\mathsf{fma}\left(\cos delta, \sin \phi_1, \left(\sin delta \cdot \cos \phi_1\right) \cdot \cos theta\right)\right) \cdot \sin \sin^{-1} \left(\mathsf{fma}\left(\cos delta, \sin \phi_1, \left(\sin delta \cdot \cos \phi_1\right) \cdot \cos theta\right)\right)\right) \cdot \sin \sin^{-1} \left(\mathsf{fma}\left(\cos delta, \sin \phi_1, \left(\sin delta \cdot \cos \phi_1\right) \cdot \cos theta\right)\right)\right)}\right)}}} \]

  7. Simplified0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\sqrt[3]{\color{blue}{{\left(\cos delta - \sqrt[3]{{\sin \sin^{-1} \left(\mathsf{fma}\left(\cos delta, \sin \phi_1, \sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right)\right)}^{3} \cdot {\sin \phi_1}^{3}}\right)}^{3}}}} \]

  8. Applied flip3--_binary640.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\sqrt[3]{{\color{blue}{\left(\frac{{\cos delta}^{3} - {\left(\sqrt[3]{{\sin \sin^{-1} \left(\mathsf{fma}\left(\cos delta, \sin \phi_1, \sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right)\right)}^{3} \cdot {\sin \phi_1}^{3}}\right)}^{3}}{\cos delta \cdot \cos delta + \left(\sqrt[3]{{\sin \sin^{-1} \left(\mathsf{fma}\left(\cos delta, \sin \phi_1, \sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right)\right)}^{3} \cdot {\sin \phi_1}^{3}} \cdot \sqrt[3]{{\sin \sin^{-1} \left(\mathsf{fma}\left(\cos delta, \sin \phi_1, \sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right)\right)}^{3} \cdot {\sin \phi_1}^{3}} + \cos delta \cdot \sqrt[3]{{\sin \sin^{-1} \left(\mathsf{fma}\left(\cos delta, \sin \phi_1, \sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right)\right)}^{3} \cdot {\sin \phi_1}^{3}}\right)}\right)}}^{3}}} \]

  9. Applied cube-div_binary640.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\sqrt[3]{\color{blue}{\frac{{\left({\cos delta}^{3} - {\left(\sqrt[3]{{\sin \sin^{-1} \left(\mathsf{fma}\left(\cos delta, \sin \phi_1, \sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right)\right)}^{3} \cdot {\sin \phi_1}^{3}}\right)}^{3}\right)}^{3}}{{\left(\cos delta \cdot \cos delta + \left(\sqrt[3]{{\sin \sin^{-1} \left(\mathsf{fma}\left(\cos delta, \sin \phi_1, \sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right)\right)}^{3} \cdot {\sin \phi_1}^{3}} \cdot \sqrt[3]{{\sin \sin^{-1} \left(\mathsf{fma}\left(\cos delta, \sin \phi_1, \sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right)\right)}^{3} \cdot {\sin \phi_1}^{3}} + \cos delta \cdot \sqrt[3]{{\sin \sin^{-1} \left(\mathsf{fma}\left(\cos delta, \sin \phi_1, \sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right)\right)}^{3} \cdot {\sin \phi_1}^{3}}\right)\right)}^{3}}}}} \]

  10. Simplified0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\sqrt[3]{\frac{\color{blue}{{\left({\cos delta}^{3} - {\sin \sin^{-1} \left(\mathsf{fma}\left(\cos delta, \sin \phi_1, \sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right)\right)}^{3} \cdot {\sin \phi_1}^{3}\right)}^{3}}}{{\left(\cos delta \cdot \cos delta + \left(\sqrt[3]{{\sin \sin^{-1} \left(\mathsf{fma}\left(\cos delta, \sin \phi_1, \sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right)\right)}^{3} \cdot {\sin \phi_1}^{3}} \cdot \sqrt[3]{{\sin \sin^{-1} \left(\mathsf{fma}\left(\cos delta, \sin \phi_1, \sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right)\right)}^{3} \cdot {\sin \phi_1}^{3}} + \cos delta \cdot \sqrt[3]{{\sin \sin^{-1} \left(\mathsf{fma}\left(\cos delta, \sin \phi_1, \sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right)\right)}^{3} \cdot {\sin \phi_1}^{3}}\right)\right)}^{3}}}} \]

  11. Simplified0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\sqrt[3]{\frac{{\left({\cos delta}^{3} - {\sin \sin^{-1} \left(\mathsf{fma}\left(\cos delta, \sin \phi_1, \sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right)\right)}^{3} \cdot {\sin \phi_1}^{3}\right)}^{3}}{\color{blue}{{\left(\mathsf{fma}\left(\sqrt[3]{{\sin \sin^{-1} \left(\mathsf{fma}\left(\cos delta, \sin \phi_1, \sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right)\right)}^{3} \cdot {\sin \phi_1}^{3}}, \cos delta + \sqrt[3]{{\sin \sin^{-1} \left(\mathsf{fma}\left(\cos delta, \sin \phi_1, \sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right)\right)}^{3} \cdot {\sin \phi_1}^{3}}, {\cos delta}^{2}\right)\right)}^{3}}}}} \]

  12. Final simplification0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\sqrt[3]{\frac{{\left({\cos delta}^{3} - {\sin \sin^{-1} \left(\mathsf{fma}\left(\cos delta, \sin \phi_1, \sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right)\right)}^{3} \cdot {\sin \phi_1}^{3}\right)}^{3}}{{\left(\mathsf{fma}\left(\sqrt[3]{{\sin \sin^{-1} \left(\mathsf{fma}\left(\cos delta, \sin \phi_1, \sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right)\right)}^{3} \cdot {\sin \phi_1}^{3}}, \cos delta + \sqrt[3]{{\sin \sin^{-1} \left(\mathsf{fma}\left(\cos delta, \sin \phi_1, \sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right)\right)}^{3} \cdot {\sin \phi_1}^{3}}, {\cos delta}^{2}\right)\right)}^{3}}}} \]

Reproduce

herbie shell --seed 2021224 
(FPCore (lambda1 phi1 phi2 delta theta)
  :name "Destination given bearing on a great circle"
  :precision binary64
  (+ lambda1 (atan2 (* (* (sin theta) (sin delta)) (cos phi1)) (- (cos delta) (* (sin phi1) (sin (asin (+ (* (sin phi1) (cos delta)) (* (* (cos phi1) (sin delta)) (cos theta))))))))))